On rational connectedness and parametrization of finite Galois extensions

Abstract

Given two $G$-Galois extensions of $\mathbb{Q}$, is there an extension of $\mathbb{Q}(t)$ that specializes to both? The equivalence relation on $G$-Galois extension of $\mathbb{Q}$, induced by the above question, is called $R$-equivalence. The number of $R$-equivlance classes indicates how many rational spaces are required in order to parametrize all $G$-Galois extensions of $\mathbb{Q}$. We determine the $R$-equivalence classes for basic families of groups $G$, and consequently obtain parametrizations of the $G$-Galois extensions of $\mathbb{Q}$ in the absence of a generic extension for $G$.

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